Therefore, by (ii), there exists \(d_*=d_*(n)\in {\mathbb {N}}\)  such that \(d(n)\leqslant d_*(n)\)  for all \(n\) , \(d=o(d_*)\)  (in particular, \(d_*\rightarrow \infty \) ), and (ii) holds for \(d\)  replaced by \(d_*\) . As shown above, the latter implies that (i) holds for \(d\)  replaced by \(d_*\) . By Maclaurin's inequality (see (12.3) in [1]}),
\(\frac{S_n}{n}-1\geqslant \ln \frac{S_n}{n}\geqslant \frac{1}{d} \ln \frac{S_n^{(d)}}{\binom{n}{d}} \geqslant \frac{1}{d_*} \ln \frac{S_n^{(d_*)}}{\binom{n}{d_*}} ,\) 
